Title: Dirac Operators in Riemannian Geometry
Speaker: Yongqiang Tian（Central South University）
Time: Monday, October 17, 2022, 14:00-15:00
Location: B201-1, Mingde Building
Abstract：As is well known, the Dirac operator plays a crucial role in Alain Connes’ noncommutative geometry. In this talk, we will revisit the construction of Dirac operators on Riemannian spin manifolds. Some basic knowledge of classical geometry is required.
Working on a spectral triple (A,H,D), i.e. the noncommutative generalization of a Riemannian spin manifold, places you into the operator framework. So, in order to get some non-trivial results on it, both summability and regularity concerning the abstract Dirac operator D (self-adjoint, possibly unbounded) are usually assumed to be good enough. However, life is not smooth, especially when your algebra A is not ‘smooth’ either. Suppose now we have a nice algebra acting on Hilbert space H, then how to construct a proper Dirac operator D to guarantee the summability and regularity? There is no routine method in the noncommutative setting. This motivates us to look for some inspirations from the starting point: Riemannian geometry! And we will provide a few examples.
More information: Graduate Student Seminar
